Solutions to Dirichlet problem on manifolds with boundary

For ball in $\mathbb{R^n}$ it is easy. You can cover whole ball with one map(for example identity map :D) And write you equation $\Delta w =0$ in coordinates. This will only alter the Laplace operator a little bit and you end up solving equation in to form $$ \nabla \cdot (A\,\nabla w) = 0 $$ Where $$ A_{ij} = \sqrt{|g|} g^{ij} $$ see wiki page on Laplace-Beltrami operator.

So you have standard Dirichlet problem problem in $\mathbb{R^n}$, therefore existence and uniqueness is guaranteed. If you want reference that for example consult Evans book on PDE.

On any compact manifold with boundary. I think that the result should be the same, in the proof existence on $\mathbb{R^n}$ you do not use any global properties of $\mathbb{R^n}$. I think It should be reasonably easy to alter the existence proof from $\mathbb{R^n}$ to any compact manifold with boundary, but a little bit tedious because you have to develop theory of Sobolev spaces on that manifold.

On compact manifold without boundary. Here I answer a question that has not been asked but seems to me interesting.

What about equation $$ \Delta w = g $$

Here we have to be careful. Necessary condition on existence is $\int g = 0$. This can be seen from a little physical intuition. This equation can be considered as stationary solution of heat equation $$ \partial_t w - \Delta w = g $$ In stationary situation there has to be balance between energy inflow and outflow. Without boundary the cannot be any energy flow through it, thus the necessary condition $\int g = 0$. Although I do not know if it is sufficient condition for existence.

But it is probably very similar to the purely Neumann problem, where condition $\int g=0$ is sufficient.